3 problems

Let $G$ be a $k$-connected graph on $n$ vertices with minimum degree $\delta(G)\ge k+2$. A matching is $k$-removable when deleting its edges leaves a $k$-connected graph. The maxim…

For $k\ge 1$, a vertex set $S$ in a $k$-connected graph $G$ is $k$-removable when $G-S$ remains $k$-connected; a matching is $k$-removable when its edge deletion leaves a $k$-conne…

For $k\ge 1$, let $T$ be a tree of order $m$, and let $G$ be a $k$-connected or $k$-edge-connected graph with minimum degree $\delta(G)\ge k+m-1$. A subtree $T'$ is isomorphic to…